UKPSC Prelims 2024 General Studies Paper-I · Question 89 of 147

Correct — B, 8 months.

Half-life is the time in which half of whatever is present decays, and it applies afresh to what remains. After 4 months, 1/2 of the substance is left. After another 4 months, half of that half — 1/4 — is left.

If 1/4 remains, then 3/4 has decayed. That takes two half-lives, so the answer is 8 months.

The idea to carry away: convert the fraction decayed into the fraction remaining, then count halvings. 3/4 decayed means 1/4 left, and 1/4 = (1/2)², so two half-lives: 2 × 4 = 8 months.

• (a)
  12 months — Twelve months is three half-lives. After three halvings, 1/8 remains and 7/8 has decayed, which is more than the 3/4 the stem asks about.

  Twelve months is the right answer to how long for 7/8 of the substance to decay with a 4-month half-life.

• (c)
  4 months — Four months is one half-life. After it, 1/2 remains and 1/2 has decayed, so the process the stem describes is only half done.

  Four months is the right answer to how long for half of the substance to decay, which is the definition of half-life itself.

• (d)
  3 months — Three months is 4 × 3/4, the half-life scaled by the fraction decayed. That treats decay as linear, as if a fixed amount goes each month.

  Radioactive decay is exponential: each interval removes a fixed fraction of what is left. Three months is under one half-life, so less than half has decayed by then.

  Three months is the right answer to a linear-rate problem in which a fixed 1/4 of the original amount decays each month, which is not how radioactive decay proceeds.

Half-life (T½) is the time in which the quantity of a radioactive nuclide falls to half its starting value. Decay is exponential: in each half-life, half of whatever remains decays, so the same T½ applies at every stage regardless of the amount present.

After n half-lives the fraction remaining is (1/2)ⁿ and the fraction decayed is 1 − (1/2)ⁿ; the elapsed time is n × T½. Half-life is tied to the decay constant λ by T½ = ln 2 / λ ≈ 0.693/λ.

• Half-life is the time for half of a radioactive sample to decay and is a constant characteristic of the nuclide.
• After 1, 2, 3 and 4 half-lives the fraction remaining is 1/2, 1/4, 1/8 and 1/16 respectively.
• A fraction decayed of 3/4 means 1/4 remains, which is (1/2)², i.e. two half-lives.
• Half-life and decay constant are related by T½ = ln 2 / λ ≈ 0.693/λ.
• Mean life τ = 1/λ is longer than the half-life: τ ≈ 1.44 × T½.
• Half-life is practically independent of temperature, pressure and the chemical state of the nuclide.
• Carbon-14 has a half-life of about 5,730 years and is the basis of radiocarbon dating.
• Rutherford and Soddy put forward the exponential law of radioactive transformation in 1902–03.

• Reading 'decay 3/4' as 'remain 3/4': the fraction gone must be converted to the fraction left before counting half-lives.
• Scaling the half-life linearly (4 × 3/4 = 3 months) as if a fixed amount decays each month; each half-life removes half of what is left, not half of the original.
• Counting three halvings for 3/4 because the numerator is 3; 3/4 decayed is 1/4 left, which is two halvings, not three.
• Stopping at one half-life because 'half' appears in the problem; one half-life leaves 1/2, not 1/4.

• practice — not a real PYQ

  The half-life of a radioactive isotope is 20 days. What fraction of the original sample will remain after 60 days?

  • (a)1/2
  • (b)1/4
  • (c)1/8
  • (d)1/16
  Answerc — 60 days is 60 ÷ 20 = 3 half-lives, and (1/2)³ = 1/8 remains.

  Option (a) 1/2 is what remains after one half-life (20 days); (b) 1/4 after two half-lives (40 days); (d) 1/16 after four half-lives (80 days). Only (c) matches three halvings.

• practice — not a real PYQ

  A radioactive sample decays to 1/16 of its initial quantity in 2 hours. Its half-life is:

  • (a)15 minutes
  • (b)30 minutes
  • (c)45 minutes
  • (d)60 minutes
  Answerb — 1/16 = (1/2)⁴, so four half-lives fit into 120 minutes, giving 30 minutes each.

  With (a) 15 minutes, 120 minutes would be eight half-lives and 1/256 would remain. With (c) 45 minutes, 120 ÷ 45 is not a whole number of half-lives and about 1/6 would remain. With (d) 60 minutes, only two half-lives pass and 1/4 remains.

• practice — not a real PYQ

  87.5% of a radioactive sample has decayed in 12 years. The half-life of the sample is:

  • (a)3 years
  • (b)4 years
  • (c)6 years
  • (d)12 years
  Answerb — 87.5% decayed means 12.5% = 1/8 remains, which is (1/2)³, three half-lives in 12 years, so 4 years each.

  Option (a) 3 years would mean four half-lives in 12 years and 1/16 remaining (93.75% decayed). Option (c) 6 years gives two half-lives and 75% decayed. Option (d) 12 years gives one half-life and 50% decayed.

• practice — not a real PYQ

  Which of the following statements about the half-life of a radioactive nuclide is correct?

  • (a)It doubles when the initial amount of the sample is doubled
  • (b)It is the time in which the entire sample decays
  • (c)It is practically independent of temperature and pressure
  • (d)It is equal to the mean life of the nuclide
  Answerc — half-life is a nuclear property, practically unaffected by temperature, pressure or chemical combination.

  Option (a) fails because half-life is independent of the amount present; a larger sample still halves in the same time. Option (b) fails because half-life is the time for half the sample to decay, not all of it. Option (d) fails because mean life is 1/λ, about 1.44 times the half-life.